This means that we can make the solution as small as
want without leaving the feasible set S and so this is a Unbounded case solution

2) The Vector yj = B-1aj has someone positive component

Them is possible to build other feasible solution (it will be into the extreme point of S) where the function is smaller

The new solutions is build as follows
x = w + λdj

being dj = (-B-1aj ej)t
and
λ = min{β/yij : yij > 0}, β = B-1 b.

With this value λ, x is feasible solution and if for example

λ = βj/ yij
Then x is the basic solution associed to the matrix

B, = (a1, a2, ..., ar-1, aj, ar+1, ..., , am )

Ie,
That is, we changed a vector by another (we have substituted the vector in r position for which is in place j).

In resume we have built a solution from other.
Note: if there are various indexes satisfying

zj - cj > 0

Then the operations is performed on the index k that satisfies

zj - ck = max (zj - cj), con zj - cj > 0

Overview of the Simplex Method

The extreme points x0, ..., xs, are the solutions for the systms

Bx=b, with B mxm submatrix of initial matrix A.

The optimal solution, if there, should be between these points...

Given an extreme point xi in the feasible set, we check if the Optimality criterion holds for it, if does then xi is the optimal solution founded.

Otherwise, one of these two situations can happen:
1) Unbouded Solution.
2) It is possible to iterate to next extreme point and
test the optimality criterion for it.

This is the simplex algorithm basic idea, iterating between the extreme points of feasible set,
which are solutions of linear systems equations taken from square submatrices of constraints matrix A, and make this until
one of them meets the optimality criterion.
There remains the problem of obtaining these solutions, we will see in
section Simplex calculations
how Simplex algorithm
offers us a calculation in which there is not need to perform the inverse of the matrix A. This calculation is called pivoting the matrix and is another basic element of the simplex algorithm.